On a new type of mixed interpolation

摘要

We approximate every function f by a function fn(x) of the form a cos kx + b sin kx + Σn−2i=0cixi so that f(jh) = fn(jh) for the n + 1 equidistant points jh, j = 0,…, n. That interpolation function fn(x) is proved to be unique and can be written as the sum of the nth-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.